Electrical parameters extraction of PV modules using artificial hummingbird optimizer

The parameter extraction of PV models is a nonlinear and multi-model optimization problem. However, it is essential to correctly estimate the parameters of the PV units due to their impact on the PV system efficiency in terms of power and current production. As a result, this study introduces a developed Artificial Hummingbird Technique (AHT) to generate the best values of the ungiven parameters of these PV units. The AHT mimics hummingbirds' unique flying abilities and foraging methods in the wild. The AHT is compared with numerous recent inspired techniques which are tuna swarm optimizer, African vulture’s optimizer, teaching learning studying-based optimizer and other recent optimization techniques. The statistical studies and experimental findings show that AHT outperforms other methods in extracting the parameters of various PV models of STM6-40/36, KC200GT and PWP 201 polycrystalline. The AHT’s performance is evaluated using the datasheet provided by the manufacturer. To highlight the AHT dominance, its performance is compared to those of other competing techniques. The simulation outcomes demonstrate that the AHT algorithm features a quick processing time and steadily convergence in consort with keeping an elevated level of accuracy in the offered solution.

Solar energy is a promising renewable technology due to its environmental responsiveness and numerous supplies. Solar Photovoltaic (PV) system development is continuing, which encourages the effective use of these systems in generating electric power to meet the need for energy 1 . Also, there are several drawbacks to the performance of PV systems, such as insufficient PV panel productivity and direct panel disclosure to the elements 2 . As a result, determining the realistic efficiency of PV systems is critical for efficiently planning, controlling, and simulating PV modules. To achieve this goal, the practical model is employed based on the current and voltage samples that are gathered at the module terminals. PV parameters may be established, and its model can be built with the aid of mathematical representation.
In the literature, many researchers have developed a variety of PV models, including the Single-Diode Model (SDM) and Double-Diode Model (DDM). Furthermore, PV model performance depends on unidentified internal parameters. Due to degradation, aging, and unpredictable functioning states, keeping all the unknown parameters steady and evaluating them is challenging. Designing, estimating, simulating, and optimizing PV modules is impossible without establishing their regarding electrical parameters. As a result, the effectiveness of swarm optimizing methods for quantifying PV system parameters is being studied 3 . Analytical approaches 4 create simplified assumptions or particular approximations with ignoring compromising accuracy. However, this analytical model has been simplified by ignoring the effect of parallel and series resistances in calculating the current and voltage related to the highest power output. In 5 , Lagrange Multiplier Method (LMM) have been proposed for SDM/DDM to optimize power outputs of solar cell PV modules. In 6 , the crucial information were reduced from the datasheet of manufacturer where a bounding requirement for a zero-voltage state has been created using the power first derivative. Furthermore, in 7 , four random locations have been illustrated on the I-V curve and their slopes to extract the SDM parameters analytically without approximation or simplification. However, such analytical approach is limited to conventional testing scenarios. Which has a lot of calculations and fails when they change 8 .
On the other side, numerical approaches including deterministic and metaheuristic algorithms have been presented. Inaccurate initial values might lead to local optima in the deterministic method as well as the real model finds it challenging to satisfy the objective function equation's limitations 9 . Conversely, metaheuristic methods SDM. This configuration typically contains of one diode (D) in parallel with shunt resistor ( R sh ) and photogenerated current ( I ph ). This configuration, as manifested in Fig. 1, is put in series with another ( R S ). The SDM current output (I) is mathematically formulate 36,37 : where I P , I D1 , and I SD1 denote the current through shunt resistor, diode current, the diode's reverse saturating current, accordingly. Moreover, the symbol ( n 1 ) characterizes its ideality factor, while the symbol (V) demonstrates the output voltage. In addition to that the symbol ( V tm ) indicates the thermal voltage, whereas the electron charge (q) value is C, the Boltzmann constant ( K b ) value is J/K, and T refers to the cell temperature. In this context, it is important to define five parameters f(I ph , R S , n, I SD , R sh ) accurately from (1).

DDM.
It is considered as a SDM with insertion of an additional recombination diode (D2) as manifested with aid of Fig. 2. Although SDM has advantages of its simple structure, fewer parameters to extract, and rapid implementation in the depletion area, It ignores the recombination loss at low voltage which is pivotal when using the (1) Objective Function. The objective function can diminish the error among simulated and experimental current by defining the optimal estimation of the electrical parameters of the two models (SDM and DDM). The root mean square error (RMSE) 40 is used as an objective function to determine the change between two I-V characteristics. It can be illustrated as follows: where M explains the number of experimental data points; I j exp and V j exp represent the current and voltage values of j th experimental point, respectively; I j cal V j exp , x denotes the computed current output; and the variable x implies the decision parameters.
The objective presented in Eq. (4a) is so traditional and implemented by various reported works. Based on this objective model, the main focus is for minimizing the error aggregation, but it doesn't guarantee the direction for minimizing the maximum error which may be produced via a single experimental recording. Despite the objective model in Eq. (4a) provides significant coincidence for the whole PV characteristics, the error in some readings would be high. Therefore, another objective function is dedicated to minimize the summation of current absolute error (MAE) over the number of experimental data points which can be mathematically modeled as follows:  www.nature.com/scientificreports/ Using the proposed objective in Eq. (4b), the searching direction is dedicated for minimizing the maximum error over the course of the experimental recordings and so the distribution of errors will be approximately equivalent and more suitable.

AHT for extraction of PV cell parameters
In the AHT's procedures, each hummingbird is assigned a definite food supply from which it can be feed. For this specific feed supply, it can memorize the rate and location of nectar replenishment. Moreover, it remembers how often it has been since it last accessed every source of food. The AHT has exceptional capability for finding the best solutions thanks to these special skills. Initially, a swarm with h n size of hummingbirds, as indicated in (5), is randomly assigned to h n food sources: where H i refers to the location of the ith food source which depicts the PV cells parameters as a solution vector. The upper and lower bounds with problem dimension are demonstrated by Ub and Lb ; and R describes a randomized vector between [0, 1]. A visitation table (VT) of the food sources is established using the following criteria: where VT i,k shows how many times the ith hummingbird failed to reach the kth source of food, and null denotes the absence of any value. Three flight maneuvers-axial, diagonal, and omnidirectional flights-are extensively used throughout foraging and modelled in the AHT which can each be seen in (7) as follows: where r1 denotes a randomly generated value falling inside [0, 1]; rand i and randperm denotes randomized generating functions to create values in the form of integers and permuted integers, correspondingly.
The directed and territorial tactics of the hummingbirds randomly choose one of the flight skills described in (7). At first, a hummingbird uses the directed foraging method to inspect its intended source of food, which results in the discovery of a potential feed ingredient which is explained by: where H i (t) and H i,t arg et (t) represent the positions of the present and intended i th food sources at time t; N(0, 1) is the Gaussian distribution function.
Secondly, based on a certain territorial parameter (b), the territorial foraging strategy involves searching for a fresher food supply inside the surrounding area, as shown below: Therefore, the location of every food source (i) can be generally updated by: where O (·) expresses the objective target described in (4). According to this model.
Third is the migratory foraging technique, in which hummingbirds would frequently fly into a more distant location to get foods whenever their region is food-scarce 41 . The hummingbirds could travel to different feeding sources, selected at random from the whole seeking area.
where H worst denotes the source that has the minimum nectar replenishment rate within the population.
An inspection process should be performed ensuring that each hummingbird is always travelling inside the boundary searching space, and consequently each dimension variable that was abused, according to (14), will be returned to the search space boundary:  (6).
The importance of this visiting table is to keep track of the time each food supply has passed without being revisited by a single bird, with a long gap between visits indicating a higher visit rate. Thus, Fig. 3 exhibits the principal steps of the AHT in recognizing the unidentified parameters of PV cells.  To compare the suggested AHT with some other newly established methods when used on the SDM and DDM of STM 6-40/36, four effectiveness metrics of maximum, average, minimum and standard deviation of the RMSE are displayed. Additionally, a low RMSE score shows that parameters were obtained effectively since RMSE aims to minimize the difference between measured and simulated data. Additionally, the AHT is validated on the PWP 201 polycrystalline PV module utilizing both SDM and DDM. The simulations are carried out with MATLAB 2017b on Intel® Core™ i7-7500U CPU @ 2.70 GHz 2.90 GHz with 8.00 GB RAM. Two scenarios are considered based on the selected objective function as follows: • Scenario 1: Traditional objective model presented in Eq. (4a) for minimizing the error aggregation.  Table 2. In terms of the numerical simulations, for SDM, the proposed AHT could achieve the lowest possible value of 1.7298E−3, whilst AVO, TSO, and TLSBO achieve the lowest possible values of 1.7324E−3, 1.9219E−3, and 1.9264E−3, respectively as manifested in Table 2. The AHT could achieve the lowest possible value of 1.7028E−3, whilst AVO, and TSO achieve the lowest possible values of 1.7049E−3 and 2.6843E−3, accordingly for DDM.
In addition, thirty independent runs are conducted for the proposed AHT, AVO, TSO, and TLSBO for SDM and DDM, in this article, to show the performance of these optimizers. It can be noticed from conducting these runs that the proposed AHT has the minimum value among these techniques which highlight the efficiency and robustness of the proposed AHT compared with these optimizers as exemplified in Fig. 4. As shown, for SDM, AHT derives the least minimum, mean, maximum and standard deviations related to the RMSE of 0.001729814, 0.001729831, 0.001730045 and 5.39233E−8, respectively. On contrary, AVO, TLSBO and TSO obtain higher standard deviations of 0.001084, 0.000102 and 0.000967, respectively. Similar findings are acquired for DDM, AHT achieves the least minimum, mean, maximum and standard deviations related to the RMSE of 0.001704932, Table 1. Bounded ranges per cell of the PV parameters.  www.nature.com/scientificreports/ 0.001728661, 0.001762892 and 9.85118E−6, respectively. On contrary, AVO, TLSBO and TSO, respectively, obtain higher standard deviations of 0.000833907, 0.000537932 and 0.000899015. The convergence characteristics of AHT are developed for SDM and DDM as illustrated in Fig. 5a,b and compared to AVO, TSO, TLSBO, Forensic-Based Investigation (FBI) Technique 44 , Enhanced marine predator approach (EMPA) 45 , Equilibrium Optimization (EO), Heap-based Technique 29,46,47 , and Jellyfish search (JFS) optimizer 48 . It can be manifested from this figure that the convergence characteristics of the AHT has an excellent performance contrasting to these optimizers.
For SDM, the experimental and estimated data illustrated by AHT, AVO, TSO, and TLSBO of the currents and powers are pronounced in Fig. 6a,b, respectively for each 20 points. Besides, at each point, the absolute error between estimated and experimental data for the currents and powers are exemplified as manifested in the previously mentioned figures, where the proposed AHT achieves the lowest absolute errors in comparison with    Fig. 6a, the maximum absolute percentage errors between measured and estimated current values is 0.14% at the experimental point no. 11 with IAE value of less than 6.09 mA cropped by the AHT. The I-V curve and the P-V curve are illustrated in Fig. 7a,b, respectively which exemplifies the precise proximity between both the estimated and experimental data of the powers and currents at each point of voltage. For DDM, the experimental data and the estimated data illustrated by AHT, AVO, TSO, and TLSBO of the currents and powers are described in Fig. 8(a)-(b), respectively for each 20 points. Besides, at each point, the     Tables 3 and 4, correspondingly. In these tables, the proposed AHT is compared to several optimizers of AVO, TSO and TLSBO, and the reported optimizers of simulated annealing (SA) 49 , three point based approach (TPBA) 50 , hybridizing cuckoo search/biogeography based optimization (BHCS) 51 , ITLBO 25 , enhanced CS approach (ECSA) 52 , improved shuffled complex evolution (ISCE) 53 , chaotic logistic Rao technique (CLRT) 54 , enhanced PSO (EPSO) 55 , fractional chaotic PSO (FC-EPSO) 56 , bat optimization approach (BA), novel BA (NBA), directional bat algorithm (DBA) 57 , SDO 58 , MPA 59 and improved chaotic whale optimization algorithm (ICWOA) 61 for both models.
As demonstrated, the AHT achieves the least RMSE, standard deviation, mean and maximum of 1.7298E−3, 5.3923E−8, 1.7298E−3, 1.7300E−3, respectively, for SDM (see Table 3). The AHT achieves 1.7049E−3, 9.8512E−6, 1.7287E−3, and 1.7629E−3, respectively for DDM as indicated in Table 4. The comparative assessment exemplifies high search accuracy and good stability of the suggested AHT compared to several newly techniques and the reported optimizers.   www.nature.com/scientificreports/ Table 5. Also, the convergence characteristics of the AHT in comparison with AVO, TLSBO, and TSO for this scenario are developed for SDM and DDM as illustrated in Fig. 10a,b. In terms of the numerical simulations, for SDM, the proposed AHT could achieve the lowest MAE value of 4.068E−3, whilst AVO, TSO, and TLSBO achieve the lowest possible values of 8.805E−3, 6.175E−3 and 6.193E−3, respectively as manifested in Table 5.     www.nature.com/scientificreports/ For both models at this scenario, the experimental absolute errors in the produced current are described in Fig. 12a,b for the AHT, AVO, TLSBO, and TSO, respectively for each 20 points. As shown, the proposed AHT derives superior capability compared to the others in minimizing the maximal absolute error. Based on this scenario, the distribution of errors is approximately equivalent and more suitable where the searching direction is dedicated for minimizing the maximum error over the course of the experimental recordings. For the SDM, the errors using the proposed AHT range from 0.000495 at the reading no. 7 to 0.004901 at the reading no. 11. Therefore, the regarding difference between the two obtained boundaries is 0.004406. In similar way, the calculated difference between the two obtained boundaries using AVO, TLSBO, and TSO are 0.02237, 0.005984 and 0.006012, respectively. These differences demonstrate the high capability of the proposed AHT in achieving the best distribution of the errors over the course of the experimental recordings.
For the DDM, the errors using the proposed AHT range from 0.000389 at the reading no. 18 to 0.004415 at the reading no. 11. The regarding difference between the two obtained boundaries is 0.00403. In similar way, the calculated difference between the two obtained boundaries using AVO, TLSBO, and TSO are 0.01452, 0.007083 and 0.007797, respectively. These differences demonstrate the high capability of the proposed AHT in achieving the best distribution of the errors over the course of the experimental recordings.   Table 6. Besides, for SDM, the proposed AHT could achieve the lowest possible value of 6.4957E−4, whilst AVO, TSO, and TLSBO achieve the lowest possible values of 1.0426E−2, 1.1538E−2, and 1.2897E−2, respectively. It is also seen from this table that the proposed AHT could achieve the lowest possible value of 3.7154E−4, whilst AVO, TSO, and TLSBO achieve the lowest possible values of 9.6100E−3, 1.1268E−2 and 1.2580E−2, respectively for DDM. Thirty independent runs are conducted for the proposed AHT, AVO, TSO, and TLSBO for SDM and DDM of this module, in this article, to show the performance of these optimizers. It can be noticed from conducting these runs that the proposed AHT has the minimum value among these techniques which highlight the efficiency and robustness of the proposed AHT compared with these optimizers as exemplified in Fig. 13.
As shown, for SDM, AHT acquires the least minimum, mean, maximum and standard deviations related to the RMSE of 0.0006496, 0.0067283, 0.0095589 and 0.0023414, respectively. On contrary, AVO, TLSBO and TSO obtain higher mean RMSE of 0.037613, 0.015379 and 0.041981, respectively. As well, they obtain higher standard deviations of 0.009989, 0.00121 and 0.011769, respectively. Similar findings are acquired for DDM, AHT obtains the least minimum, mean, maximum and standard deviations related to the RMSE of 0.000371545, 0.009679481, 0.016820415 and 0.005198108, respectively. On contrary, AVO, TLSBO and TSO, respectively, obtain higher standard deviations of 0.011457421, 0.001713595 and 0.013369672.
The convergence characteristics of AHT for SDM and DDM are illustrated in Fig. 14a,b, respectively and compared to AVO, TSO, TLSBO, EMPA, EO, Heap, JFS and FBI 44 . It can be manifested from this figure that the convergence characteristics of the AHT has an excellent performance in comparison with these optimizers.
For SDM, the experimental data and the estimated data illustrated by AHT, AVO, TSO, and TLSBO of the currents and powers are described in Fig. 15a,b, respectively for each 15 points. Besides, at each point, the absolute error between estimated and experimental data for the currents and powers are exemplified as manifested, where the proposed AHT achieves the lowest absolute errors in comparison with AVO, TSO, and TLSBO. For sake of quantifications, as shown in Fig. 15, the maximum absolute percentage errors between measured and estimated current values is 0.45% at the experimental point no. 9 with IAE value of 11.2 mA cropped by AHT.  www.nature.com/scientificreports/ The I-V curve and the P-V curve are illustrated in Fig. 16a,b which exemplifies the closeness between both the estimated and experimental data of the powers and currents at each point of voltage. For SDM, the experimental data and the estimated data illustrated by AHT, AVO, TSO, and TLSBO of the currents and powers are described in Fig. 17a,b, respectively for each 15 points. Besides, at each point, the absolute error between estimated and experimental data for the currents and powers are exemplified, where the proposed AHT achieves the lowest absolute errors in comparison with AVO, TSO, and TLSBO. The I-V curve and the P-V curve are illustrated in Fig. 18a,b which exemplifies the closeness between both the estimated and experimental data of the powers and currents at each point of voltage.
For the KC200GT PV module, the statistical analysis for SDM and DDM are demonstrated in Tables 7 and 8 Table 7. On the other hand, the AHT achieves 3.7154E−4, 5.1981E−3, 9.6795E−3, and 1.6820E−2, respectively for DDM (See Table 8). The comparative assessment exemplifies high search accuracy and good stability of the proposed AHT in comparison with the recently developed optimizers and the reported optimizers.  Table 7. Comparative assessment of the compared optimizers for KC200GT PV module using SDM (Scenario 1).  Table 8. Comparative assessment of the compared optimizers for KC200GT PV module using DDM (Scenario 1).  Figs. 19a,b and 17a,b provide the corresponding I-V and P-V curves for KC200GT PV module. With temperature variations at irradiance level of 1000 W/m 2 , as shown in Fig. 19a,b, the proposed AHT derives significant coincidence between the simulated and experimental recordings. Similar findings are obtained with irradiance variations at temperature of 25 °C as shown in Fig. 20a,b. Both figures indicate the high validation of the proposed AHT at different temperatures and solar radiations.

Optimizer Max (RMSE) Mean (RMSE) Min (RMSE) Std (RMSE) Population number Maximum number of iterations
Simulated results of scenario 2 for KC200GT PV module. For this scenario, the proposed AHT, AVO, TSO, and TLSBO are performed and the regarding parameters of SDM and DDM are depicted in Table 9. Also, the convergence characteristics of the AHT in comparison with AVO, TLSBO, and TSO for this scenario are developed for SDM and DDM as illustrated in Fig. 21a,b. As shown, for SDM, the proposed AHT could achieve the lowest    Fig. 22 displays the regarding Whisker's plot of the AHT in comparison with AVO, TLSBO, and TSO for this scenario.
As shown, for SDM, AHT derives the least minimum, mean, maximum and standard deviation related to the MAE of 0.0279, 0.0488, 0.0589 and 0.0092, respectively. On the other side, AVO, TLSBO and TSO obtain higher standard deviations of 0.0183, 0.0133 and 0.0136, respectively. Similarly, for the DDM, the proposed AHT shows the best performance with the least minimum, mean and maximum MAE values of 0.017, 0.0349 and 0.0647, respectively.
For both models at this scenario, Fig. 23a,b describe the experimental absolute errors in the produced current for the AHT, AVO, TLSBO, and TSO, respectively. As shown, the proposed AHT derives superior capability with better distribution of errors compared to the others. For the SDM, the errors using the proposed AHT range from 0.0026 at the reading no. 3 to 0.0279 at the reading no. 12. Therefore, the regarding difference between the two obtained boundaries is 0.0253. In similar way, the calculated difference between the two obtained boundaries using AVO, TLSBO, and TSO are 0.0333, 0.0313 and 0.0388, respectively. These differences demonstrate the high capability of the proposed AHT in achieving the best distribution of the errors over the course of the experimental recordings.
For the DDM, the errors using the proposed AHT range from 4.18E−5 at the reading no. 3 to 0.01701 at the reading no. 6. The regarding difference between the two obtained boundaries is 0.001697. In similar way, the calculated difference between the two obtained boundaries using AVO, TLSBO, and TSO are 0.01963, 0.03101 and 0.01874, respectively. These differences demonstrate the high capability of the proposed AHT in achieving the best distribution of the errors over the course of the experimental recordings.
PHOTO WATT-PWP 201 PV module. Simulated results of scenario 1 for PWP 201 PV module. Table 10 describes the five and seven-nine variables of SDM and DDM, respectively, that were obtained using the AHT considering both scenarios 1 and 2. According to Table 10, for the first scenario, the AHT approach determines that 2.42507 mA is the optimum adaption value for both SDM and DDM. The minimal RMSE associated with    Fig. 24 illustrates the AHT's convergence properties which demonstrates how the suggested AHT shows high performances for both SDM and DDM. Additionally, Fig. 25a-d exhibits the projected and measured values for the powers and currents at each point of the SDM and DDM of this module, characterizing the similarity between the anticipated and measured values while estimating the data with the proposed AHT.   Table 10. For both models and scenarios, Fig. 26a,b display the experimental absolute errors in the produced current using the proposed AHT. For the SDM, in the first scenario, the errors using the proposed AHT range from 9.21E−5 to 4.43E−3 with regarding difference between the two obtained boundaries of 4.34E−3. For the same model, the errors using the proposed AHT range from 4.40E−5 to 3.66E−3 with regarding difference between the two obtained boundaries of 3.62E−3 considering the second scenario. Based on that, the utilization of the MAE minimization objective at Scenario 2 shows better error distribution with 16.67% improvement over the RMSE minimization objective at Scenario 1 via the proposed AHT. Similar findings are attained considering the SDM. The utilization of the MAE minimization objective at Scenario 2 shows better error distribution with 15.15% improvement over the RMSE minimization objective at Scenario 1 via the proposed AHT. In the first scenario, the errors using the proposed AHT range from 9.22E−5 to 4.43E−3 with regarding difference between the two obtained boundaries of 4.34E−3. For the same model, the errors using the proposed AHT range from 2.9E−5 to 3.71-3 with regarding difference between the two obtained boundaries of 3.68E−3 considering the second scenario.   www.nature.com/scientificreports/ Moreover, a comparative assessment between the proposed AHT, AVO, TSO, and TLSBO is performed considering this scenario. The regarding parameters of SDM and DDM are depicted in Table 12 while their convergence characteristics are developed for SDM and DDM as illustrated in Fig. 27a,b. As shown, the proposed AHT could achieve the lowest MAE value of 3.66E−3 and 3.71E−3, for SDM and DDM, respectively.
In addition, Fig. 28 displays the regarding Whisker's plot of the AHT in comparison with AVO, TLSBO, and TSO for this scenario. As shown, the AHT shows the best performance compared to the others. The proposed AHT derives the least minimum, mean, and maximum related to the MAE of 0.00366, 0.00451 and 0.00628 for the SDM and 0.00371, 0.005696 and 0.01286 for the DDM, respectively.

Conclusion
This study has presented a novel application of an Artificial Hummingbird Technique (AHT) for extracting the unknown parameters from SDM and DDM PV models of mono-crystalline STM6-40/36, and multi-crystalline KC200GT. The performance of the proposed AHT is assessed by statical indices called Min RMSE, Max RMSE, Mean RMSE, Standard deviation, IAE, PAE, P-V and I-V curves. The earlier results of AHT for determining accurate parameters of various PV models illustrate that AHT produces a competitive end-result versus other recently developed algorithms. The parameters of the PV module are extracted using the AHT in this article.
To estimate the PV module parameters, the proposed approach uses experimental data extracted from the Power-Voltage (P-V) curve. At a final stage of this effort, Photo WATT-PWP 201 has been examined. To sum up, three distinct PV modules, which are widely used in the literature namely, STM6-40/36, KC200GT and Photo WATT-PWP 201 have been investigated to validate the proposed AHT. For all PV modules, the proposed AHT exhibits the lowest RMSE. The performance of the AHT is additionally tested utilizing statistical data overall 30 independent runs. Based on the experimental results, it may be announced that the AHT overcomes all the selected state-of-the-art optimizers for the reported test cases.

Data avaliability
The data that support the findings of this study are available from the corresponding author upon reasonable request.